On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

Gerlach, E.; Meichsner, Jan; Skokos, Ch.

doi:10.1140/epjst/e2016-02657-0

Cited by 12 publications

(17 citation statements)

References 46 publications

(61 reference statements)

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“…As a result, using the same computational resources and CPU times one can integrate the KG models for times of one or two orders of magnitude higher than the times achieved for the DNLS systems. For this reason ways to improve the efficiency of the symplectic integration of the DNLS model were investigated in [47,48] where the performance of several SIs based on three part splits, with different orders of accuracy, were studied in detail.…”

Section: Introductionmentioning

confidence: 99%

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Computational efficiency of symplectic integration schemes: application to multidimensional disordered Klein–Gordon lattices

Senyange

Skokos

2018

Eur. Phys. J. Spec. Top.

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We implement several symplectic integrators, which are based on two part splitting, for studying the chaotic behavior of one-and two-dimensional disordered Klein-Gordon lattices with many degrees of freedom and investigate their numerical performance. For this purpose, we perform extensive numerical simulations by considering many different initial energy excitations and following the evolution of the created wave packets in the various dynamical regimes exhibited by these models. We compare the efficiency of the considered integrators by checking their ability to correctly reproduce several features of the wave packets propagation, like the characteristics of the created energy distribution and the time evolution of the systems' maximum Lyapunov exponent estimator. Among the tested integrators the fourth order ABA864 scheme [59] showed the best performance as it needed the least CPU time for capturing the correct dynamical behavior of all considered cases when a moderate accuracy in conserving the systems' total energy value was required. Among the higher order schemes used to achieve a better accuracy in the energy conservation, the sixth order scheme s11ABA82 6 exhibited the best performance.

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Section: Introductionmentioning

confidence: 99%

“…For all these reasons SIs based on splitting the system's Hamiltonian in distinct, integrable parts have already been used for the integration of the KG and DNLS models [12,15,17,18,23,26,30,31,34,47,48,38]. In these studies SIs belonging mainly in the so-called SABA schemes [49] were used.…”

Section: Introductionmentioning

confidence: 99%

Computational efficiency of symplectic integration schemes: application to multidimensional disordered Klein–Gordon lattices

Senyange

Skokos

2018

Eur. Phys. J. Spec. Top.

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“…. , N are N constants of motion, and the Hamiltonian functions of the q− and p−hoppings We note that a rather thorough survey on three part split SIs can be found in [49,50]. We decided to include in our study a smaller number of schemes than the one presented in these works, focusing on the more efficient SIs.…”

Section: Three Part Splitmentioning

confidence: 99%

“…[KL] in [49,50]) s17ABC8 having 69 individual steps s17ABC8(τ) = ABC2(δ 1 τ)ABC2(δ 2 τ) × · · · × ABC2(δ 8 τ)ABC2(δ 9 τ)ABC2(δ 8 τ) × · · · × ABC2(δ 2 τ)ABC2(δ 1 τ).…”

Section: Three Part Splitunclassified

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Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

Danieli¹,

Manda²,

Mithun³

et al. 2019

Mathematics in Engineering

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We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the Fermi-Pasta-Ulam-Tsingou (FPUT) chain and the one-and two-dimensional disordered, discrete nonlinear Schrödinger equations (DDNLS). In our analysis we consider methods based on Taylor series expansion, Runge-Kutta discretization and symplectic transformations. The latter have the ability to exactly preserve the symplectic structure of Hamiltonian systems, which results in keeping bounded the error of the system's computed total energy. We perform extensive numerical simulations for several initial conditions of the studied models and compare the numerical efficiency of the used integrators by testing their ability to accurately reproduce characteristics of the systems' dynamics and quantify their chaoticity through the computation of the maximum Lyapunov exponent. We also report the expressions of the implemented symplectic schemes and provide the explicit forms of the used differential operators. Among the tested numerical schemes the symplectic integrators ABA864 and S RKN a 14 exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models s9ABC6 and s11ABC6 (moderate accuracy), along with s17ABC8 and s19ABC8 (high accuracy) proved to be the most efficient schemes.are the 2N × 2N elements of the Hessian matrix D 2 H (x(t)) of the Hamiltonian function H computed on the phase space trajectory x(t). Eq. (3.5) is linear in w(t), with coefficients depending on the system's trajectory x(t). Therefore, one has to integrate the variational equations (3.5) along with the equations of motion (3.2), which means to evolve in time the general vector X(t) = (x(t), δx(t)) by solving the systemẊH (x(t)) · δx(t).( 3.7) In what follows we will briefly describe several numerical schemes for integrating the set of equations (3.7).

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Theory of localization-hindered thermalization in nonlinear multimode photonics

Ramos¹,

Shi²,

Fernández-Alcázar³

et al. 2023

Commun Phys

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Our society’s appetite for ultra-high bandwidth communication networks and high-power optical sources, together with recent breakthroughs in mode multiplexing/demultiplexing schemes, forced the photonics community to reconsider the deployment of nonlinear multimode systems. These developments pose fundamental challenges stemming from the complexity of nonlinear mode-mode mixing by which they exchange energy in the process towards an equilibrium Rayleigh-Jeans (RJ) distribution. Here we develop a universal one-parameter scaling theory for the relaxation rates of out-of-equilibrium excitations towards their RJ thermal state. The theory predicts an exponential suppression of the rates with increasing disorder due to the formation of stable localization clusters resisting the nonlinear mode-mode interactions that tend to separate them. For low optical temperatures, the rates experience a crossover from linear to nonlinear temperature dependence which reflects a disorder-induced reorganization of the low frequency eigenmodes. Our theory will guide the design of nonlinear multimode photonic networks with tailored relaxation-scales.

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On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

Cited by 12 publications

References 46 publications

Computational efficiency of symplectic integration schemes: application to multidimensional disordered Klein–Gordon lattices

Computational efficiency of symplectic integration schemes: application to multidimensional disordered Klein–Gordon lattices

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

Theory of localization-hindered thermalization in nonlinear multimode photonics

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